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Theorem speccl 26482
Description: The spectrum of an operator is a set of complex numbers. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)
Assertion
Ref Expression
speccl  |-  ( T : ~H --> ~H  ->  (
Lambda `  T )  C_  CC )

Proof of Theorem speccl
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 specval 26481 . 2  |-  ( T : ~H --> ~H  ->  (
Lambda `  T )  =  { x  e.  CC  |  -.  ( T  -op  ( x  .op  (  _I  |`  ~H ) ) ) : ~H -1-1-> ~H }
)
2 ssrab2 3580 . 2  |-  { x  e.  CC  |  -.  ( T  -op  ( x  .op  (  _I  |`  ~H )
) ) : ~H -1-1-> ~H }  C_  CC
31, 2syl6eqss 3549 1  |-  ( T : ~H --> ~H  ->  (
Lambda `  T )  C_  CC )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   {crab 2813    C_ wss 3471    _I cid 4785    |` cres 4996   -->wf 5577   -1-1->wf1 5578   ` cfv 5581  (class class class)co 6277   CCcc 9481   ~Hchil 25500    .op chot 25520    -op chod 25521   Lambdacspc 25542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-hilex 25580
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-map 7414  df-spec 26438
This theorem is referenced by: (None)
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